On the Onsager-Machlup functional for elliptic diffusion processes

Abstract

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Construction of the landscape for multi-stable systems: Potential landscape, quasi-potential, A-type integral and beyond

“…and the orthogonality condition (x) · ∇φ QP (x) = 0 (28) holds by (25). We note here that the decomposition (27) is the zero noise limit of (6) because εφ P L (x) → φ QP (x), ε∇ · D(x) → 0 and J ss (x)/P ss (x) → (x) by the decomposition equality of b(x).…”

“…For the state-dependent diffusion matrix D(x), it is argued in [25] that the term 1 2 ∇ · b(ψ) in (10) should be replaced by [26], it is claimed that an additional term involving second order derivative of σ(x) (only the one dimensional case is considered in [26]) should be added in order to give the correct solution for Ito-type Fokker-Planck Equation. We remark that the general mathematical expression for OM function in high dimensional cases has been studied in [27,28], which include the result in [26] as a special case.…”

Asymptotic evaluation of the Poisson measures for tubes around jump curves

“…We refer to Ikeda and Watanabe (1981) and Shepp and Zeitouni (1992) for some basic results in that direction, and to Lyons and Zeitouni (1999) and Capitaine (2000) for theorems concerning the consistency of the OnsagerMachlup functional with respect to the norm considered on the Wiener space. Note also that the case of SDEs in infinite dimensions driven by a Gaussian noise have been considered in Mayer-Wolf and Zeitouni (1993), and Tindel (2000, 2001).…”

Tube estimates for diffusion processes under a weak Hörmander condition

“…In their work · is the Euclidean norm, but later on different norms have been used to take into account the regularity of the trajectories (about this, see for example [9] and [20]). This problem is interesting for physicists because of the Onsager-Machlup functional (see [24], [13]), and is also related to large and moderate deviation theory (see [11], [21]).…”